Zeqiri, 21, arrived from Swiss Super League side Lausanne-Sport on undisclosed terms, reported to be £3.6m.
The striker has represented Switzerland through the country’s junior levels from under-16s to under-21s and agreed a contract with Albion until June 2024.
Last season he scored 22 times in 37 appearances in all competitions, as Giorgio Contini’s side were promoted as champions to the top-tier of Swiss football.
The 6ft 1in forward had looked set to join Parma and there was also interest from Wolfsburg, Nice and Marseille but Brighton's playing style and their reputation for developing young talent convinced the Kosovan Swiss forward.
Albion head coach Graham Potter has given young talented players first team chances and helped launch the Premier League careers of the likes of Aaron Connolly, Steven Alzate, Alexis Mac Allister and Tariq Lamptey.
“I know he puts faith in young players," said Zeqiri who joined Lausanne in 2011 and made his senior debut a month before his 16th birthday.
"He has a good way of playing, which I like. I know they often have the ball and they like to have a lot of possession.
“I had the opportunity to see what the coach was proposing to me in terms of the playing system. It is honestly something that really convinced me – I hope to do great things with him.
“Brighton are a good team. I’ve seen them play a few times, especially against United which was really good and I could really see myself as part of that.
"I hope we can do some good things this season.”
Zeqiri had a season-long loan with the Juventus academy in 2016, and returned to his hometown club and became a regular in the Lausanne first team.
The striker is confident he can replicate that form in the Premier League.
He said, “I’m a forward who really likes playing with my back to goal. I’ve got a powerful shot and I am good finisher. I can play in a few positions.”